Length and area partitioning methods and apparatus



Jan. 24, 1967 LENGTH AND AREA PARTITIONING METHODS AND APPARATUS 'w. F. GALEY ET AL Original Filed Nov. 2, 1959 11 Sheets-Sheet 1 GENERAL METHOD STEPS DETERMINING AVAILABLE WIDTHS FOR EACH PROGRAMMED LENGTH FROM LEADING EDGE ACCORDING TO DEFECT LOCATIONS.

FITTING PROGRAMMED WIDTHS FOR EACH PROGRAMMED LENGTH INTO AVAILABLE WlDTI-IS CHOOSING PROGRAMMED LENGTH HAVING BEST WIDTH FIT DETERMINING AVAILABLE WIDTHS FOR EACH PROGRAMMED LENGTH FROM NEW LEADING EDGE CUTTING FIRST CHOSEN FITTING PROGRAMMED WIDTHS FOR EACH PROGRAIIMED LENGTH INTO AVAILABLE WIDTHS PROGRAMMED NEW LENGTH LEADING EDGE CUTTING FITTED WIDTH-S OF FIRST CHOSEN LENGTH CHOOSING PROGRAMMED LENGTH HAVING BEST WIDTH FIT REPEAT STEPS CUTTING SECOND CHOSEN PROG RAMNED L ENGTH CUTTING FITTED WIDTHS OF SECOND CHOSEN LENGTH INVENTORS WILLIAM F. GALEY JOSEPH A.GULOTTA FORREST KUMBEL A? TYS,

1967 w. F. GALEY ET AL 3,300,629

LENGTH AND AREA PARTITIONING METHODS AND APPARATUS Original Filed Nov. 2, 1959 11 Sheets-Sheet 2 LEADlNG DRECTION OF GLASS EDGE Egg-es UNITSS Z UNITS B EXAMPLE OF DEFECT PATTERN PROGRAMMED 5125s DEFECT 2 s S\ZE Z 5 A zA UsE g 9 6 z e c 25 I! 9X8 8 l 8 25 43 ZOXIO 0 3 o 16 2 20 u 2 u 42 2on3 vs I I3 27 41 7 3 7 d g 2 35 ms 35 :5 2 :5 29 4 35x29 29 I Z9 Z9 40 INVENTORS Z? WILLIAM F. GALEY 3| 3s JOSEPH A.Gu|.o1"m e as 53 FORREST kflmeeg. f 44 35 6g: W0 gw/ W7 3 45 as ATTYS Jan. 24, 1967 w. F. GALEY ET 3,300,629

LENGTH AND AREA PARTITIONING METHODS AND APPARATUS Original Filed Nov. 2, 1959 ll Sheets-Sheet 3 .ZC FIRST CUT FROM LEAIZNNG EDGE 2 12 TRIAL TRlAL TRIAL AVAILABLE sms Accum- CHOSEN 1 s WIUTHSLNZGER uumao mm THAN ANY S VALUE. l 35 7,529 l6 \(IS) 55 47 9 \(1) Z 20 |3,||,|o Zl \(u),|(|o) 24 |(13),|(u) 525 38 I5 ms) 1'. 3 9 8,1, Zl 3(6) 5 1.4 3(8) 56 m If: |(e),|(e)

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(CHOSEN) b INVENTORS WILLIAM F. GAL-E! JOSEPH A.GUL.01'TA FORREST k. UHBEL b3: M/ 9%., Hu -$1M,

Jan. 24, 1967 w. F. GALEY ET AL 3,300,629

LENGTH AND AREA PARTITIONING METHODS AND APPARATUS Original Filed Nov. 2, 1959 ll Sheets-Sheet 4 1. 3 GENERAL PARTHTIONING of A=n,a,+na+na+--na E3 E2! E21 E21 :3- IZ- E g- 1- E53 1 E 3 E21 E21 i: EEI J III ll v ELI SELECTIVE PARTITIONING of A=n,a,+n,a +n a ----n a FOR A=n. T0 2 mcLusnvE a; 2,3,4 7- a RANK= 4, 5, z 6- E L 5 INVENTORS 2 WILLIAM F. GALE! JosEPH A. Guwrm Faanssr K, Umssm.

Jan. 24, 1967 w. F. GALEY ET AL 3,300,629

LENGTH AND AREA PARTITIONING METHODS AND APPARATUS Original Filed Nov. 2, 1959 ll Sheets-Sheet 5 3' c m] g Hl-n 8 a 5 0% m n v o v) v m 0 3 5w INvEN'roRs J 3 WILLIAM EGALEY LO JOSEPH A-GUL Foancsr KUMBEL. [L b WJ QL, MMfl/GJWQM A-rws Jan; 24, 1967 Original Filed Nov. 2, 1959 W. F. GALEY ET AL 11 Sheets-Sheet 6 LEGENDS 6 -g 5% 76 is z 77 E E Y M Jr +8 -1 1r [3] if H 491" I 2] 1H". 3] E 1. +6 1 I 3 1,! w w w El +4 3 W w E 3 +3 H 1 1 Z 3 H INVENTORS WILLIAM EGALEV JO$EPH A.GULOT1'A Fonnzs'r K. Umasn.

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LENGTH AND AREA PARTITIONING METHODS AND APPARATUS Original Filed Nov. 2, 1959 ll Sheets-Sheet 9 To To VOLTAGE l 3 LOWER DISTRIBUTOR oz I02 RANK amzcks n2 [08a '12 H I 02: 4 BLOCK a 5 BLOCK WILLIAM F. GALEY JosEPH A.Gu|..o'r1'A FORREST K. UMBE w. F- GALEY ETAL 3,300,629

LENGTH AND AREA PARTITIONING METHODS AND APPARATUS l1 Sheets-Sheet 10 Jan. 24, 1967 Original Filed Nov. 2, 1959 Jan. 24, 1967 w, GALEY ET AL LENGTH AND AREA PARTITIONING METHODS AND APPARATUS 11 Sheets-Sheet 11 Original Filed Nov. 2, 1959 QM O- W71 Add .rumumo F RRESTKUMBEL 7 M4GWATTYS United States Patent 3,300,629 LENGTH AND AREA PARTITIONING METHODS AND APPARATUS William F. Galey, Saxonburg, Joseph A. Gulotta, New

Kensington, and Forrest K. Umbel, Verona, Pa., assignors to Pittsburgh Plate Glass Company, Allegheny County, Pa., a corporation of Pennsylvania Original application Nov. 2, 1959, Ser. No. 850,360, now Patent No. 3,246,550, dated Apr. 19, 1966. Divided and this application Oct. 15, 1964, Ser. No. 411,958

15 Claims. (Cl. 235-185) This application is a division of our copending application Serial No. 850,360, filed November 2, 1959, now Patent No. 3,246,550.

This invention relates to the fitting of given lengths or areas into given total lengths or areas, typically in terms of partitioning sheet material having random defect areas into predetermined defect-free sizes. It has particular reference to cutting plate glass into sale-able sizes after inspection in a continuous glass manufacturing process.

Plate glass, as a sheet material, is presented for inspection after it has been ground, polished and washed, and prior to cutting it into various marketable sizes. The glass is subject to defects of various types and unpredictable size and location which may make out sheets containing the defect areas totally unusable or which may affect the quality grading of sale-able sheets. In fitting or piecing out different combinations of the various sizes from the usable areas it is desired, of course, to throw away or downgrade as little as is necessary of the glass which is free of defects of a severity affecting the glass grade being cut. Time and timing are always of concern since the operation is an adjunct of high production manufacture in which glass of given width, either as large separate sheets, or sometimes as an unbroken ribbon, moves at a continuous rate as it is formed and finished.

Efiiciency goals in laying out the grade-defined defectfree area of the total glass output into usable sizes have in fact been limited by the number of factors which skilled layout men have found it practical to consider. The possible factors are many. For example, a large number of predetermined sizes are marketable and more than one saleable grade may also be involved. Inspection is required for the total output since the defect distribution is not predictable. Relative preference or demand for the various sizes may also be factors. The demand for particular sizes may change in accordance with the quantities which have already been cut. Obviously, the number of choices of possible glass sizes which must be evaluated before a cut is made is very large if all of the factors or choices are considered.

Human judgment under such conditions is often intuitive at best, making it at least ditficult, and always more expensive, to reach decisions quicker by multiplying manpower. Yet complex problems usually require time for their full solution, and the economies of manufacture make the cost of the time involved part of the definition of the least wasteful cutting decision. In the interests of reducing storage and handling of the glass produced before shipping, close control of elapsed time is also involved. Desirably the sequence of inspection for defects, the presentation of all the relevant data, the making of economic cutting decisions, the actual cutting, and the routing of the cut glass sizes to respective packing stations, is an essentially on-line procedure. The present invention makes this possible.

It is, therefore, a principal object of the invention to provide methods and apparatus for making intelligently the decisions involved in cutting a large sheet into desired smaller sizes. Stated in another way, it is an object of the invention to provide methods and apparatus for correlating all supplied data on specific defect locations in a sheet to be cut on the one hand, and on desired sizes on the other hand, to reach timely, systematic, and economic cutting decisions without resort to human judgment. Another object is to provide a method and means for systematically employing defect location data and trial dimensions in laying out defect-free rectangles.

It is an object to provide a computer for automatically relating defect data and demand data in reaching cutting decisions for sheet material such as plate glass.

It is also an object to provide computer circuits for quickly and reliably determining choices available and selecting the best choice according to a programmed logic. Further, it is an object to provide a computer which is readily incorporated on a continuous glass production line.

Other objects include provision of methods and apparatus for converting the output of a plate glass factory into saleable glass sizes of maximum value or with minimum waste. It is also an objective to provide automatic means for determining both length and area partitions of any total length or area. A further objective is to provide apparatus for partitioning sheet glass which is relatively fast, simple, and inexpensive.

Summarizing, it is an objective to simultaneously remove defects, meet quantity and size demands, and reduce waste all within the bounds of new but economically available technology.

In addition to the partitioning of glass or other sheet materials, it will be appreciated that the lengths or area quantities are represented generally by integers to be partitioned into combinations of numbers representing members of a set of partitioning element numbers.

It is therefore a general object to provide methods and apparatus for partitioning.

Further, it is an object to provide means for determining either general or selective integral partitions for any integral quantity.

It is another object to provide digitally exact partitioning without resort to binary arithmetic systems.

It is also an object to provide means for computing partitions without requiring read-out of previously stored solutions.

In terms of computer apparatus, it is a further object to provide partitioning computers having logic circuits minimizing storage and programming of arithmetic steps. It is also an object to provide a computer apparatus operable at reasonable speeds with relatively simple and inexpensive electromagnetic switch components.

Other features, objects and advantages of the invention will become apparent from the following detailed description illustrated in the accompanying drawings in which:

FIGURE 1 outlines the steps of a two-cut logic in piecing out sheet material in accordance with the invention.

FIG. 2 is an example of a leading portion of a glass sheet with particular defect pattern located thereon.

FIG. 2A is a table of the defect pattern example locating the defective unit squares of FIG. 2.

FIG. 2B is a table of exemplary programmed sizes with rank and value to be cut from the glass sheet of FIG. 2.

FIG. 2C is a table illustrating the trial partitions of the sheet of FIG. 2 leading to choice of a particular set of cutting commands according to the program-med sizes of FIG. 2B.

FIGS. 2D, 2E, and 2F graphically portray the trial partitions that establish the table of FIG. 2C.

FIG. 3 is a graphical representation of a general solution to a partitioning problem.

FIG. 4 is a graphical representation of a selective solution to a partitioning problem.

FIG. 5 illustrates an electrical permission matrix analog of the partitioning definition and example of FIG. 3.

FIG. 6 illustrates an electrical command matrix analog of the selective partitioning definition example of FIG. 4.

FIG. 7 is a circuit diagram of a size switch and readout indicator circuit for the matrix of FIG. 6.

FIG. 8A represents a 12-unit length of a sheet bearing defects in the first and ninth units.

FIG. 8B is a diagram of a 12-unit defect relay circuit for reporting the defects of FIG. 8A.

FIG. 8C is a circuit diagram of a series-type volt-age distributor operated by the defect relays of FIG. 8B for energizing the matrix of FIG. 6.

FIG. 8D represents the voltage connections to the matrix made by the series-type voltage distributor of FIG. 8C.

FIG. 8B is a circuit diagram of a parallel type voltage distributor operated by the defect relays of FIG. 8B for energizing the matrix of FIG. 6.

FIG. 8F represents the voltage connections to the matrix made by the voltage distributor Oi FIG. 8E.

FIG. 9 is a diagram of a universal cell block in a 12- span matrix like that of FIG. 6.

FIG. 10 is a diagram of the cell size selector switch and the read-out indicator for the matrix cell block of FIG. 9.

FIG. 11 illustrates the cell relay connection of each cell in the block of FIG. 9.

FIG. 12 graphically indicates the connection of 3 universal cell blocks shown in FIG. 9 as a selective matrix for a, integers of 4, 3, and 2 in that rank.

FIG. 13 illustrates the connection of two complementary cell blocks as a parallel return dual matrix for effectively applying rated coil voltage to all cell relays in partitioning current paths and with exemplary a, solutions of 4 and 3.

FIG. 14 illustrates a universal cell-block dual matrix as arranged for selection of integers of 3 through 12 and with an exemplary selection of a =4.

FIG. 15 illustrates one of the priority or read-out subassemblies of the FIG. 13 matrix.

FIG. 16 illustrates a universal cell-block switch control circuit for the FIG. 14 matrix.

FIG. 17 represents the defect switches, voltage distributor, automatic programming, and marker circuits as incorporated in a system with the matrix of FIG. 14,

Although certain particular embodiments of the invention have been shown and described here in some detail there is no intention to hereby limit the invention to the specific forms or details illustrated. On the contrary, the invention is to cover all alternatives, modifications, and equivalents falling within the spirit and scope of the invention as expressed in the appended claims, and the specification has been organized to guide the reader to that end as concisely and clearly as the complexities of some of the described industrial embodiments permit.

For an over-all view and introduction, Section I presents both the general method involved and definitive analysis of partitioning as here employed. Section II introduces physical matrices as analogs of partitioning equations together with the construction and operation of several useful circuits for practicing the invention.

I. PRACT ICING THE METHOD OF THE INVENTION A. Area partitioning The over-all method of area partitioning of sheet material in accordance with the invention involves what may be termed a two-cut logic. For a generalized representation of the method in terms of the steps involved, the reader is referred to FIG. 1 which relates to plate glass as the sheet material out of which defect-free prescribed sizes are to the pieced.

Preliminarily, in cutting plate glass as a sheet material, it should be noted that three basic cutting requirements are respected.

First, the sheet or continuous ribbon is to be partitioned into rectangles of prescheduled or programmed dimensions, as distinguished from an operation in which the defective area is merely cut out of the sheet or in which the largest available defect-free piece of sheet material is always cut. The scheduled sizes are typically very much larger than the rectangular defect-bearing remainders which must be discarded.

Second, each cut must run completely across a dimension of the glass piece involved. This cutting requirement is inherent in the nature of glass since the cutting process is one in which the glass is scored and snapped. The same condition is significant for other sheet materials to the extent that the time and apparatus expense associate-d with cuts of partial lengths or cuts which turn corners are unduly high.

Third, the piecing or partitioning must be selective so that the scrap is of minimum area consistent with the choice of sizes into which the defect-free glass can be partitioned. Since defect severity afiects grading, it should be noted that a glass area may be defect-free as to one grade, but not as to another.

The glass sheet, as described in connection with an industrial apparatus embodiment may, for example, be quarter-inch plate produced as a continuous ribbon over ten feet wide at a rate in the vicinity of two hundred linea'l inches per minute. A traverse or cross cut across the glass ribbon width is the first cut of each operation, here called the Z-cut, and a second of second cuts, here called S-cuts, slit or slice the cut-off Z piece into smaller widths and eliminates the defects. This is the two-cut logic; the manner in which it is advantageously and uniquely applied rfollows.

Returning now to FIG. 1, the information required in piecing out the glass is the location of the leading edge of the glass, the locations of the defects, and the program of sizes to be cut.

For locating the defects, various means may be employed, suitably either visual or photo-electric inspection of the glass sheet of the over-all area to be considered before the first cut is made. This information can be considered as stored directly in or on the glass, but may be more readily recalled by addition of marks or markers. Preferably, however, the defect is secondarily stored, as on a tape or other record containing the dimensional location of each defect square with respect to the leading edge and the side edge.

The total demand program includes both the absolute sizes and the relative demand for them as reflected by assigned values per size. As to the sizes themselves, the lengths and widths of the various rectangles to be cut are arranged to group the respective width dimensions sharing a common length dimension. All of the length dimensions are thus grouped, each associated with one or more width dimensions. The relative demand for the difierent length by width sizes is preferably determined in a two-part analysis. First, a rank or order of priority is established for the respective width dimensions associated with each length dimension and second, a numerical value is established for each size so that values of groups of sizes can be accumulated and compared.

In accordance with the two-cut logic, the trials for the first cut correspond to the different programmed lengths as measured from the leading edge of the glass. Thus for each programmed length a trial length of the glass sheet is considered for which either all or parts of the total ribbon width are available for partitioning as defect-free spans, depending upon the number and locations of the defects affecting the grade being considered. Each defect is contained in a defect strip as long as the programmed length being considered.

Next the programmed widths for that particular programmed length are fitted into the available widths according to an assigned rank. The waste, which includes unfitted portions of the defect-free widths as well as the defect strips themselves, is minimized by choosing the number and variety of programmed widths that best partition the available defect-free widths. For each programmed length and corresponding widths thus tried, the values of the programmed sizes utilized are accumulated as the value associated with that trial length or cross cut.

The steps of fitting the programmed Widths into available defect-free widths and subsequent valu accumulations, is repeated for each programmed length before any cutting decision is made. In effect, a number of trial first cuts across the Width of the glass are made and by this logical process the cut for which the best fit is provided, i.e., that for which the value of the fitted programmed sizes is highest, is chosen. After the chosen cut determination has been made, commands for cutting that programmed length and subsequently cutting the fitted widths of that programmed chosen length are issued. A large number of programmed lengths or trial cross cuts may thus be considered and whether the first chosen cut is the smallest or the largest or an intermediate one of the programmed lengths depends upon the particular defects and cutting program. It will be recognized, of course, that as larger dimensioned cross cuts (i.e., programmed lengths) are considered, larger numbers of defects are probable. The values of larger sizes, however, may counterbalance the larger absolute waste per cut to provide in the long run the most efficient or least wasteful operation consistent with meeting production requirements.

After the location of the new leading edge is thus systematically determined the steps are repeated. Thus each programmed length is again considered but with respect to the new leading edge. The defect locations with respect to the leading edge are changed, of course, and the available widths are re-determined. After again fitting the programmed widths into the available Widths for each programmed length and accumulating the value of the sizes which would be cut, the second programmed length representing the best value is selected as the second cross cut in the repeated procedure. The widths for that second cut are cut or slitted and the fitting procedure is initiated again with respect to the newly created leading edge. The process continues a decision at a time for the length of the sheet material. If the overall length dimension is very large, as when a continuous ribbon of sheet material is presented, the steps are continuously repeated.

Programming is usually established in terms of daily or weekly production, and a running count is desirably made of sizes cut but not ordinarily stocked in order that the program may be revised when the quantity requirements are met. The programming should, of course, be adjusted to take into account current economic factors and demands.

It may be pointed out that while the sequence is ordered, the actual time at which the cutting occurs may be deferred. Thus a sheet may be marked for the subsequent cutting of successive chosen lengths (with the accompanying fitted widths), each chosen length representing the leading edge for the next.

In some instances, the length of the sheet material may not be very long with respect to the various programmed lengths so that an appreciable remaining length might be wasted at the end of the sheet. A further step is then preferably added by which no programmed length is considered which, if chosen, would render the remaining length of sheet unpartitionable. In this form the twoout logic requires a separate determination of length partitioning as permission to consider the first cut length.

The Z-at-a-time logic is also susceptible of extension as a basic step in a multiple Z-at-a-time system. A greater variety of choices under the same Z and S size and value program is obtained by selecting the highest value composite Z in a given length with respect to a leading edge. Each composite Z is a given number of 6 programmed Zs in a particular order with respect to the leading edge, and its value is subject to variation by the order in which its component Z sizes are arranged. This calls for separately evaluating the different permutations of the component Z lengths for a given composite Z. The information storage requirements are very high in such a system where substantial numbers of Z by S sizes are considered, and the preferred apparatus systems specifically described herein operate on the basic Z-at-a-time logic.

B. Specific example of glass area partitioning An example of a defect-spotted glass ribbon is shown in FIG. 2. For the purpose of usefully relating the defect locations to given desired defect-free sizes, a basic dimensional unit is established for which all of the length and width requirements are integral multiples. With a ribbon 124 inches wide as in FIG. 2, a unit of 2 inches has been chosen, giving the glass an effective width dimension of 62 units (S dimension). Of course, if the programmed sizes for cutting Were required to vary by as little as 1 inch intervals, the unit size would be selected as 1 inch. However, the unit square remains much smaller than the programmed sizes as a first step in reducing the amount of defect-free glass which must be discarded with the actual defect areas.

Considering the sheet as a mosaic or matrix of 2-inch squares, the defects are located by the Z and S coordinates for the unit square in which they lie. The glass is measured in units back from the leading edge along the length or Z-dimension, and the width or S dimensions are measured from one of the side edges. As illustrated, the coordinates apply to the squares themselves. Their boundaries are thereby indicated. By considering the beginning boundary edge of the first Z or S strip as zero, the number of the squares is the further Z or S boundary or limit line. Thus a square Z6, S22 is bounded by Z unit dimension lines 5 and 6 and by S unit dimension lines 21 and 22.

Looking again to FIG. 2, a number of defects spotted on the glass are located by Z and S coordinates in the table of FIG. 2A. The defect areas are intended to be random since in actual production their number and location cannot be predicted. The defects may be microscopic in size, occupying much less than a unit square, or they may cut across several of the mosaic units, such as the illustrated scratch defect d, but in any event, each defect is considered as occupying one square. Likewise since each square must be examined, either in whole -or in part according to the nature of the defects involved to determine its defect classification, the minimum unit dimension may be effected by the limitations of the defect detecting means. While bulb edges at the sides of the ribbon are, in eifect, continuous lengthwise defects, such predictable defects are simply eliminated by trimming. Accordingly, the locating edges herein referred to are reference axes or lines existing as physical edges after any predetermined trimming or squaring cuts are made, whether or not such cuts are deferred until such time as defect measurements or the size cuts are made.

The table of FIG. 2B illustrates a simplified program of sizes to be cut in which eight sizes are listed which may be reduced to three different length or Z-dimensions each having a group of two or three Width of S-dimensions, associated with them. Thus for any given Z-cut along the length of the glass sheet there may be a choice of Scuts. In the table of FIG. 2B, the larger of each S in a family of S-dimensions for a particular Z ranks over each smaller S. Furthermore, each Z X S size is assigned a value which, in this particular case, corresponds to its S dimension of that size. This means, for the example indicated, that different areas having the same S dimension are of equal value, but this is not necessarily anomalous as will be seen.

With this information supplied as to the sizes desired and the defect distribution, the usable areas are pieced out by the two-cut logic (as shown in the table of FIG. 2C). Accordingly each of the Z dimensions (35, 20 and 9) is made the basis of a trial or tentative first cut as measured from the lea-ding edge of the glass. Then as measured from the reference side edge of the glass, each defect is marked off as a strip one unit Wide leaving particular available defect-free widths.

In fitting the S dimensions into an available width, the minimum waste partition is resolved by employing the highest ranked Ss wherever a choice occurs. If there is no exact partition, the available width is decreased by one, and the fit again tried. This process is repeated until a perfect partition is gained or the defect-free width is exhausted. FIGS. 2D, 2E, and 2F graphically portray the S-fits for the three Z trials tabulated in FIG. 2C. The cross-hatched area represents the defect strip; the singlehatch areas represent the glass discarded as cullet with the defect strip if the Z cut is elected, due to the impossi-bility of making an initial fit. Sufficiently large waste pieces may be saved, if desired, piecing out special or unprogrammed sizes.

By totalling the assigned values of the utilized sizes for each trial Z, the Z choice is dictated by the highest accumulation. In this case the Z of 20 is chosen since the value is 58only two unit defect strips plus two additional unit strips of unfitta'ble waste glass are involved.

In the event of a tie in summed values, the tie is systematically broken by initially electing to pick the first (-or the last) of tying values encountered and choosing the order of the Z trials accordingly. For this example, while no tie was present, the Z trials were made in the order of decreasing Z length and had a tie of largest value occurred it would have been broken in favor of the first totaled value, i.e., the largest Z.

With the choices of rank in making S fits, of value in choosing a Z after trial fitting, and of order of Z trials for breaking ties, practically any programming demand can 'be anticipated without requiring human judgment after the program has been set up. It will be appreciated, with the example illustrated, a minimum area Waste per Z unit length logic is dictated by making value proportional to S. If value were proportional to Z x S in dimensional units, then the Z of 35 would be chosen in this example, despite the larger amount of waste. If in the S fitting the rank is not in the order of the S dimension, ties of partitioning in the S dimension are broken in favor of the smaller Ss and they would, of course, be cut. In the long run (i.e., a days, a weeks, a years production) the waste is by definition minimal since the programmed sizes have been cut in the programmed order to meet the de mand thus analyzed.

In the case where the ribbon is cut into discrete sheets before inspection, as, for example, 180 inch (or 90 unit) sheets, left-over lengths are preferably avoided. This is done by always choosing a Z which is part of an exact fit of the various Z dimensions in the total length. If the 'highest value Z is not available, the next highest is chosen, and so on. In this instance, any of 9, 20, or 35 may be chosen as the first cut without defeating the lengthwise fit. Choice of a Z of 20 in the first cut leaves 70 units, int-o which two Z lengths of 35 may fit. However, a Z of 20 (or 9) would be eliminated from consideration for the second cut.

Usually, by filling in the program with a large number of Zs, a greater number of lengthwise choices remain open. The number of trial Zs which are subject to the S trial partitions and valuation are, of course increased, but the burden of added decisions is lightened by their still systematic resolution. If the choice is likely to be limited by reason of few or only large Z numbers programmed, a small remainder in the total length may be tolerated. A preferred method is to try alternatively partitions of the total length number as decreased, by one, two, and three, for example. A fit in any case calls for only a small waste.

C. Basic partitioning definitions At the heart of two-dimensional or area partitioning is a basic method of one-dimensional or length partitioning. Such partitioning may be generally expressed in terms of sets of integers whose sum is exactly equal to a given integer. Partitioning a large sum. into selected smaller integers, like factoring a large number into prime factors, is often easier to do than to explain. As in factoring for primes, precalculated tables may be referred to for convenience, and the results are valid whether determined by explicitly expressed logical steps or by trial and error arithmetic techniques. Since a definition of partitioning, as distinguished from factoring, does not appear to be well established, the following expression is offered as definitive at least for the purposes of the specification and claims:

In this expression A is the integer partitioned (i.e., a composite) and a, may generally refer to the integers of a set a a a etc. which fit into A without remainder. The number of each such integer 0, involved in a partition is n n n etc., or n,. The value of it, may be zero in the event that its associated a is not present in the partition. The use of the cardinal subscripts 1 2 3 need not imply any sequential or quantitative order, nor is any numerical identity to be assumed between the n, and a, to which the same subscript number is applied.

The sole partition is obvious if the only a, is the integer 1. It is equally obvious if the only a, has a value equal to A. There would be no practical problem, and no need to even term the process partitioning were only these trivial partitions involved. But this is not the general case.

Significantly, in view of the discovered utility of the partitioning concept as defined, two types of solutions are of particular interest. One may be termed the general solution in which all of the possible partitions of A by the a, integers are recognized and utilized. The other may be tenrned the selective solution in which conditions for uniqueness of solution are imposed by assigning rank or order of priority to the different given a, integers.

General partitioning is utilized in the Z partitioning method previously discussed. Whether or not a given a, is present (i.e., is its 11 other than zero?) in any of the possible partitions of A, is the information required. Information as to the number of such partitions in which that a, is present or the number of times (21,) the a is present in any one partition may not be necessary. Such a determination is not useful if the a; set includes the integer one, and higher a, integers are necessarily implied. Reference is made to FIG. 3 which shows graphically the possible fits of any or all of the a, integers 3, 5, 8, and 12, into an A integer of 20. The A is shown by an ordered array of evenly spaced parallel horizontal lines 0, 1, 2, 3, 20 (only the line segments adjacent the numbers are shown; their extensions are =unscribed for clarity of the drawing figure). The partitions are represented by vertical chains of links having respective length spacing lines differing in number by 3, 5, '8 or 12 as indicated. Thus, for an A of 20, referring to FIG. 3, links of 3, 5, 8, and 12 in various chains each shown as terminating on line 20, thus indicating that for an A of 20 any one of these a integers may be present in a partition. The choice of any of 3, 5, 8 or 12 as the first Z cut from a Z length of 20 is thus acceptable.

FIG. 3 is further arranged to show the availability of the various a integers for each of all the possible succeeding choices. Thus, if an a, of 3 is the first choice, the A becomes 17, and it will be observed that all of the a, integers still remain available as a second choice since links of 3, 5, 8 and 12 depend from line 17. If the second chosen a, is 12, a link for which extends from line 17 to line 5, then for the third choice only an a of 5 is 9 available since only a link of depends from line 5. The complete partition is thus 3, 12, 5, and the other possible partitions may be as readily traced from the table. This representation includes permutations of the sets of a integers and coefficients n constituting one partition (such It should be noted that FIG. 3 does not illustrate all partitions for the A integers less than 20, but only those reached by subtracting one or more of the a, integers. Thus no links terminate at line '16, despite the fact that it is evenly partitioned by two 8s since 16 is not a subpartition of 20.

While the general solution illustrated by FIG. 3 indicates the acceptability or future freedom of choice with respect to any selected a,, it includes the multiple choices which would make indefinite the S partitioning previously described. An assignment of rank in which, for example, a larger number always precedes, the partitions having lesser numbers of the highest a integers are eliminated. Considering the ranking a, as that a, link which is connected at its upper end to line 20 of the parallel array in FIG. 3, the highest a, of any of those available in a general partitioning of 20 is chosen. This, of course, is 12, leaving a next A of 8 for which the general partitions are 8; 5-3; or 3-5. Under the rules, 8 outranks the others, thus automatically reaching a solution for n (a which is one, and for 11 ,01 which is one 8.

FIGURE 4 further illustrates selective partitioning for all A integers up to and including 12, for a, integers of 4, 3, and 2 in that rank. The constants of the partitionequation are changed here, both to indicate the generality of the approach to a solution and provide a sutficiently simple example for which a mechanical analog can be clearly and concisely illustrated in the succeeding section. In this example it may be noted that each A is partitionable, and that there can be no ties or indecisive choice. For other examples no partition may be available, and the chosen solution is that for the next highest partitionable A. Procedurally, the A is decreased by one until a fit is found for which no remainder exists. The same result may be obtained in this particular example by setting an a, equal to 1; as a method, however, setting an at, equal to one disguises the fact that such a, may not itself be a useful or desired part of a partition. Thus where the other a, values are high, a partitioning solution may provide a large number (n,) of ones.

Since the general solution embraces the unique solution, it will be recognized that the unique or selective solution also indicates the presence of the a, integers, although in one, rather than all of the possible partitions.

II. PARTITIONING MATRICES AND ASSOCIATED CIRCUITS A. A simple matrix for general partitioning solutions A simplified electrical matrix for determining which ones of given numbers may be combined to equal a total number is illustrated in FIG. 5. It is a structural analog of the exemplary general solution of the partitioning equation illustrated in FIG. 3 and further provides partitions not only of an A of 20 but for any lesser integers selected as A. Such a matrix provides an instantaneous computation of the partitioning equation for which it is designed without resort to a storage or memory system. As a permission matrix, for example, it can determine which of several considered Z cuts from a given sheet length leave a remainder which is partitionable by the Z program.

Turning now to FIG. 5, an array of horizontal bus bars or conductors 40, numbered 0, l, 2, 3 20, represents in this instance a sequence of 20 consecutive integers. The zero bus bar, for example may represent one edge of a rectangular piece of sheet material and the successive integers represent distances along the length of the sheet as measured from that edge. The set of bus bars is 1G termed a matrix, by reason of the chains of cells or links which connect different pairs of respective bus bars. These cells 41, 42, 43, 44 correspond in this example respectively to the a, integers 3, 5, 8, and 12, each spanning a different set of bus bars spaced in number by the a, integer associated with the cell.

When a voltage which is positive with respect to the zero bus bar is applied to one of the other bars current flows downwardly through all unbroken chains of cells between the terminal bars. To prevent upward current flow, each cell contains a unidirectionally conducting or rectifying device, suitably a silicon diode 45. The diodes are poled to have a negligible voltage drop in the easy fiow direction from an upper bus to a lower one. (The contact pairs shown in each cell are later described and are closed for operation of the matrix.) It will be appreciated that the numbering of the bus bars has a structural and operational significance in view of the diode interconnections, even though the order of diode connections from left to right is a matter of choice.

Significantly, each a, bus bar span is connected by a cell of the corresponding span. The geometry selected for illustrating the circuit of FIG. 5 ties the like-span cells in chains and groups the chains in blocks to emphasize the completed interconnection. Thus the 3-span block for a =3 has three vertical chains of cells, the first starting at bus 20, the second at bus 19, and the third at bus 18. In the 12-span block the chains are limited to single cells 44 since consecutive spans would exceed the number of bus bars. All of the partitions of 20 by the set 3, 5, 8, 12, in FIG. 3 are confirmed by the presence of the corresponding a, cells of the matrix in a unidirectional current path from bus 20 to bus 0. The current path pattern of FIG. 5 is simpler than the chart of FIG. 3 in one respect because the same cell may combine current paths for several partitions. For example, a l2-span cell from bus bar 20 to bus bar 8 places a voltage on bus bar 8 for which several paths exist to the 0 bar. The FIG. 5 paths are more complex in another respect-they include permutations, and are not restricted to combinations. Thus from bus bar 8 to 0 there is one 5, 3 path through first a cell 42 and then a cell 41 and also a 3, 5 path through first a cell 41 and then a cell 42. And, of course, the FIG. 5 matrix goes beyond the diagram of FIG. 3 in that all the possible links are provided for electrical partitioning of integers less than 20.

If current flows through any of the links, the integer equal to its bus bar span is present in a partition of the total integer equal to the total energized span. Since for a permission matrix it is unnecessary to know the specific partitions, the required information is simply provided by a means responsive to current flow through only one terminal cell of each block. As shown in FIG. 5 the cell of each block which has its lower end connected to the 0 bus bar contains a read-out relay coil. As indicated, the relay coils (each shown here and in following drawings as a circle-bearing a reference numeral) for the respective blocks of 3, 5, 8 and 12 are 46, 47, 48 and 49. Other types of indicators may be employed but the relay coils have the advantage of readily providing a number of switching contacts for whatever indicating or control functions may be desired. No diode 45 is needed in the last cell containing the readout relay since no reverse paths are possible by reason of the end-of-matrix connection. All of the cells thus remain unidirectional links.

As a matter of convenience for solving partitioning equations, the respective blocks may be switched in or out of the matrix depending upon which integers are required in the set. Referring for example to the block of 3-span cells 41, a matrix size relay 50 is provided having a plurality of normally open contact pairs. This includes 17 contact pairs 50a, one in each diode cell plus another pair of 50b in the readout cell. The relay coil 50 is suitably energized by connection through a selector switch 51 to a suitable control voltage source indicated as a battery 52. With the switch 51 closed, the relay 50 is energized and all of the 3-span cells 41 are connected into the matrix. Programming of the integer 3 may be indicated as by a lamp L50 connected across the voltage source by another pair of normally opened contacts 500 of the matrix size relay.

Presence of a partition involving the integer 3 is indicated suitably by a second indicator lamp L46 connected across the control voltage source through a normally open contact pair 46a of the readout relay 46. Other control or indicating arrangements may, of course, be served by the read-out relay actuations.

Similar elements are suitably employed for the other a, blocks of 5, 8, and 12, namely, respective matrix size relays 53, 54 and 55, respective size switches 56, 57, and 58, respective size indicator lamps L53, L54, L55, and respective read-out indicator lamps L47, L48, and L49. More blocks representing other integers may likewise be switched in or out of the matrix for whatever set of integers is to be tested. Switches 51, 56, 57 and 58 are closed, of course, in the partitioning operations described.

To select the A integer to be partitioned, a positive voltage with respect to the zero bus bar is applied to the selected one of the bus bars 46. As shown, this is suitably provided as a rotary selector switch 59 having its rotor wiping the numbered bus bars and connected to the zero bus bar through the voltage source 52. With the switch connected to bus bar 20, for example, all of the programmed integers 3, 5, 8, and 12 are variously involved in some of the possible partitions of 20, and hence the read-out relay of each block will be energized. Any one of the integers is permitted i.e., its subtraction from 20 leaves a remainder which is partitionable. If, for example, a permitted cut of 12 units from a sheet 20 units long is chosen, the remainder of 8 is still partitionable by combinations involving 8, 5 and 3. This is readily determined by adjusting the selector switch 59 to the number 8 bus bar and noting that lamps L46, L47, and L48 are turned on.

Not all the As are partitionable by this integer program. For example, with the voltage connected to bus bars 7 or 4, no current path is available through any combination 3, 5, 8, and 12. And, of course, an A less than 3 is not partitionable because 3 is the smallest programmed integer. Connections of cells to all of the bus bar spans is shown, however, both as a matter of pattern consistency in illustrating the matrix and to forestall incomplete partitioning in the event other a, integers are added or substituted by switching in other blocks of cells.

If the matrix is designed to partition only a single integer some cells may be eliminated. For an A of 20, for example, and the a program illustrated, there would also be no necessity for cells connected in FIG. 5 to bus bars 19, 18, 16, 13, 7, 4, 2, or 1. These are particular rather than general cases, however, and reference to all spans as being interconnected is not intended to require connection of the unusable spans occurring because of boundary conditions or limited examples.

So long as the matrix need not identify the combinations involved in partitioning or specify the number of times a particular integer is employed in the matrix current paths, the detection or read-out of current through either terminal cell of each block of cells is sufiicient. This follows from the fact that all permutations find current paths between the terminal busses. As a result a different one of the permutations of each combination will have part of its current path through each terminal relay of each integer block anywhere involved in a current path. Read-out relays may, if desired, be provided in additional cells for securing additional information. Such modifications are illustrated more significantly in the following subsection in connection with the description of a selective partitioning matrix.

1) in B. A simple selective partitioning matrix A matrix for selective partitioning whereby a unique solution is delivered is illustrated in FIG. 6. It is a structural analog of the solution of partitioning equation illustrated by FIG. 4. As compared with a permission matrix of the FIG. 5 type, the indicated read-out integers are narrowed to only those in the highest-rank partitioning combination. Such a matrix, which also indicates the number of times each indicated integer is present may be termed a command matrix since it furnishes a particular or unique set of instructions best fitting the input conditions according to its built-in logic or decision-making instructions.

The utility of the command matrix for S partitioning may be readily realized. In addition, the FIG. 6 matrix also illustrates other useful features of partitioning matrix design for broadening its scope and versatility. Thus, since S partitioning must accommodate defects wherever they might occur, the matrix illustrated has its A spans bounded by the defects without referring the A span to the zero bus 'bar. Additionally, while the number of bus bars in the illustrated example is small, the number can be very large and several As (i.e., integeres to be partitioned) may be partitioned simultaneously.

Referring to the matrix of FIG. 6, a horizontal array of bus bars 60 and numbered 0, 1, 2, 3 12 represents a maximum A span of 12 as an analog of the selective partition example of FIG. 4.

The vertical directional characteristic of the matrix is provided as before by unidirectionally conducting cells or links 61, 62 and 63, connected respectively across bus bar spans of 2, 3, and 4. Each such cell or link contains a diode 64 poled for current flow from a higher numbered or more positive bus bar to a lower numbered bus bar. All of the available spans of 2, 3 and 4 are connected by the links which again are grouped in blocks corresponding to the a, integers of 2, 3 and 4 in the particular partitioning set here represented.

Horizontal directional characteristics establishing a declining rank from right to left are introduced by normally closed switches in the bus bars (except the zero bus bar) to the left of each block of cells. With the integers ranked 4, 3, 2 and 3-span block is to the left of the 4-span block and the 2-span block is to the left of the 3-span block. Opening a bus bar switch cuts off the positive voltage supply to all cells having their upper terminals connected to the portion of the bus bar to the left of the switch.

The over-all composite direction of rank or priority is thus from an upper right-hand positivebus bar terminal to a lower left-hand negative bus bar terminal.

Automatic operation of the bus bar switches is combined with the read-out function by providing a cell relay coil in each link. As shown the relay coils for links 61, 62 and 63 are respectively 65, 66 and 67, the coils being in series with each cell diode 64. A first normally closed contact pair 65a, 66a or 67a, associated with its respective relay coil, interrupts the bus bar to the left of the upper connection of the cell to the bus bar. A second normally open contact pair 65b, 6611, or 67b of each relay serves a suitable read-out circuit.

In operation, a positive voltage with respect to the zero bus bar applied to the positive terminal of bus bar 12 causes current flow through a path corresponding to the sole partition indicated for such a value of A by FIG. 4- namely, a chain of three 4-span cells 63 (from bus bars 12 to 8, 8 to 4 and 4 to 0) provides the sole current path. None of the other bus bars are energized, of course, and the portions of bus bars 12, 8 and 4 to the left of the 4- span block of cells 63 are disconnected by the opening of the contact pairs 67a associated with the energized relays of the current connecting cells.

If a voltage is applied between the positive terminal of bus bar 11 and the negative terminal of bus bar 2 for example, the current seeks a path across a span of 9 bus bars. With the integer program and rank here given, it will be appreciated that the correct decision must be, in order, 4, 3, 2. The sole current path is traced from *bus bar 11 through a 4-span cell 63 to bus bar 7, from bus bar 7 through a 3-span cell 62 to bus bar 4, and from bus bar 4 through a two-span cell 61 to bus bar 2. Each of the bus bars 11, 7 and 4 is interrupted by the opening of the cell relay contact pair to the left of the upper connection of each energized cell. All other current paths are thus precluded.

Matrix size relays may be desirably employed for switching different blocks of cells in and out of the matrix to program different partitioning problems. As shown in FIG. 7 matrix size relays 68, 69 and 70 are each provided with a plurality of contacts 68a, 69a and 70a which are located in the respective 2-span, 3-span and 4-span cells 61, 62 and 63 shown in FIG. 6. The relays are energized through respective size switches 71, 72 and 73, each switch and relay 'being connected across a suitable voltage source 74. All of the contact pairs are to be considered as closed, of course, in analyzing the matrix performance for the a, program of 4, 3, 2. It will be appreciated that bus bar contact pairs 65a are of significance if other blocks of cells are or may be switched into the matrix to the left of the 2-span cells 61.

Also shown in each cell or link of FIG. 6 along with the diode, the cell relay, and the matrix size contact pair is a resistor optionally added to help keep the relay current Within convenient operating limits. The resistors in each cell of a given span are of equal resistance. As shown resistors 75, 76 and 77 are provided for the cells 61, 62 and 63 respectively. The utility of the resistors and the selection of their value depends upon the voltage supply employed and is discussed in a further paragraph.

A simple read-out circuit is suitably provided by counting the number of energized cells in each block, suitably by means of normally open contact pairs from each cell relay in an indicator or control network as shown in FIG. 7. For detecting current flow through any of the Z-span cells 61, a galvanometer type indicator device 78 is connected across the voltage supply 74 through parallel circuits consisting of each cell relay normally open contact pair 65b in series with a read-out or value resistor 79. With such an indicating circuit across the voltage source 74 or other stabilized supply, the current flow through the galvanometer is zero if no 2-span cells are energized and increases with the number of contact pairs 65b which are closed.

Corresponding read-out circuits are provided for the 3-span and 4-span blocks, respectively employing galvanometers 80 and 82, read-out contact pairs 66b and 67b, and read-out resistors 81 and '83. By either adjusting the respective ratios of resistors 79, 8-1 and 83, or by providing respective galvanometer scale multiplication factors, indicated values proportional to desired multiples of the respective a integers times the number of such integers involved in a partition are obtained. Totalling the galvanometer readings thus provides a total value for the partition.

It is obvious that, without departing from the spirit of the invention, various voltage or current responsive devices, whether employed as indicators or control devices, may be substituted for the galvanometers.

Likewise, it will be appreciated that features of the matrix of FIG. 6 can be employed in a permission-type matrix. For example the normally closed contact pairs 65a, 66a and 67a in the bus bars and associated with the individual cell relays may be eliminated so that the partitioning is not selective. The galvanometer readings for the respective blocks then indicates the number of cells of each span which are energized. Embraced within this information, of course, is the indication as to whether any cell rather than no cells are involved in a partition. If only the more general information is desired, the readout network resistors 79, 81 and 83 can also be omitted. Such a modification, requiring use of cell relays in each 14 cell, is of advantage where the spans to be partitioned are not advantageously referred to the same or permanent terminal (zero) bus bar.

Whether or not it is simplified to serve as a permission matrix rather than a command matrix, the apparatus of FIG. 6, like the apparatus of FIG. 5, assures a terminal connection of at least one cell in each block. In addition it may be seen that the FIG. 6 matrix, by opening the bus bar switches, selectively excludes all but one terminal cell at each energized bus bar.

Briefly, immediately after the application of the bus bar voltage, multiple current paths are established through the matrix of FIG. 6 corresponding to all the partitioning permutations as in a permission matrix of the type shown in FIG. 5. In that sense also one cell of each block, connected either to the upper or lower bus bar of the energized span, carries current if integers of that cell span are involved in a general partitioning solution. After the energized relays open their normally closed contacts, the redundant paths are eliminated, ultimately in favor of the unique selective partitioning solution. It is an advantage of the circuit structure that relay pickup or actuation times are not critical, and the correct solution is attained even if the cell relays in the lower ranked cell spans should happen to have quicker pickup times than relays in the higher ranked spans.

C. Matrix voltage distributors While voltages may be applied by manually controlled switches across the selected bus bar span of a matrix, such as that illustrated in FIG. 6, the automatic correlation of applied matrix voltages with the location of defect-free spans of a sheet material being partitioned offers advantages for high-speed operation.

For economically meeting the requirements of a given installation, it is to some extent a matter of designers choice to either increase the number of matrix bus bars to accommodate simultaneous partitioning of several spans or apply voltage at time-spaced intervals across appropriate spans of a smaller matrix. But whether the user chooses to employ part or all of the various novel circuit structures in practicing the invention, it is not the least advantage thereof that such choices are here made available.

The voltage distributors of FIG. 8C and 8E are such novel structures. Each uses double-throw double-pole switches to apply the required voltage between related bus bars.

As a simple example FIG. 8A represents a strip of sheet material 12 units long and having defects in the first and ninth units. Lengths of seven and three units, respectively, are available for partitioning. In the defect swtiching system shown in FIG. 8B,, twelve defect relays -1 through 85-12 correspond, respectively, to the units 1 through 12 of the sheet of FIG. 8A. Relay actuators 86-1 through 86-12, shown as simple switches, energize their respective relays when closed by connecting the relay coils across a voltage supply 87. The relay contact pairs may be arranged in either a series or parallel voltage distributing system for energizing the matrix of FIG. 6 in accordance with the reported defect locations.

For the defect pattern of FIG. 8A, switches 86-1 and 86-9 are shown close-d in FIG. 83 to energize defect relays 85-1 and 85-9. Switches 86-1 and 86-9 may be, for example, automatically operated by photoelectric cells in a defect scanning circuit or manually operated in response to observation of a defect. In either event the actuated defect relay contact pairs reflect the defectfree spans beween them as well as the location of defects on the sheet. While the example is purposely simple to better illustrate complete connections to the small 12- span matrix of FIG. 6, it will be appreciated that voltage switching for much larger matrices subject to more complex defect patterns is advantageously available.

(1) A series-type voltage distributr.The series voltage distributor of FIG. 8C applies a voltage to each clear or defect-free span which is proportional to the length of the span. Twelve defect current voltage sources 88-1 through 88-12 are the unit power supplies for the respective bus bar spans corresponding to unit lengths one through twelve of the sheet of FIG. 8A. Contact pairs (1, b, c, d of the double-pole double-throw defect relays 85-1 through 85-12 connect the voltage supplies to each other and to the terminals of the FIG. 6 matrix as dictated by the defect locations. The terminals +1 through +12 and 1 through 11 in FIG. 8C, along with the zero terminal, are the ends of the bus bars of corresponding number in the FIG. 6 matrix. The switching system connects the unit power supplies in series for each defect-free span and through normally closed contact pairs 12, c. It also connects the positive and negative terminals of the series supply to the respective positive and negative bus bar terminals of the corresponding matrix bus bar span through normally open contact pairs a, d. The voltage sup-plies 88-1 and 88-9 for defect units one an-d nine are left unconnected.

Looking to the details of the FIG. 8C connection, the defect relay contact pairs in FIG. 8C are horizontally aligned with the representation of the corresponding defeet relay coil in FIG. 8B for clarity in following the Switching operations. Contact pairs at, b of each relay are a first single-pole, double-throw switch since they have a common pole or center terminal, and contact pairs 0, d of each relay are a second single pole, doublethrow switch. Referring to any unit or corresponding switch or voltage supply in terms of the number of units from the bottom of FIG. 8A, 8B, o-r 80 as In, each mth voltage supply in FIG. 8C has its negative terminal connected to the common pole of contact pairs 0, d of the m -l defect relay and its positive terminal connected to the common pole of contact pairs a, b of the m+l defect relay. The end terminals of contact pair 6 of relay in and contact pair b of relay m+1 are connected for series connection of the voltage supplies. The matrix mth bus bar connections are from the negative bus end to the end terminal of contact pair d of relay m and from the positive end to the end terminal of contact pair a of relay m+1.

End connections at the extremities of the twelve-supply series are simple. Series voltage connections are omitted, so that contact pairs a and b of defect relay 85-1 and contact pairs 0 and d of unit defect relay 85-12 are not used (and hence not shown in FIG. 8C). The positive terminal of the twelfth unit voltage supply is suitably permanently connected to the positive terminal of bus bar 12 and the negative terminal of the first unit voltage supply 88-1 is similarly connected to the zero bus bar without interfering with the selective switching function.

The operation of the voltage distributor in the absence of any defects which result in energizing the defect relays is, simple. The twelve voltage sources are simply connected in series between the positive terminal of the twelfth bus bar and the zero bus bar as would be desired, of course, in the absence of span-restricting defects. This may be readily appreciated by the fact that all of the a and d matrix c-ontact pairs remain open and all of the b and 0 series con-tact pairs closed, leaving the direct connections from the positive terminal of the twelfth volt-age supply to the positive terminal of bus bar twelve and from the negative terminal of the first voltage supply to the zero bus b'ar.

Operation under defect conditions may be appreciated by following the example of FIG. 8A in which the first and ninth units of the sheet material are defective. The circuits of FIGS. 8B and 8C accordingly show defect switches 86-1 and 86-9 closed with the contact pairs of relays 85-1 and 85-8 changed from their normal state. As a result, and as fi ther illustrated by the simplified circuit representation of FIG. 8D, voltages are applied or distributed between two different spans of the matrix. Thus, seven voltage supplies 88-2 through 88-8 are connected in series between the positive terminal of bus bar 8 and the negative terminal bus bar 1. Three voltage supplies 88-10, 88-11, and 88-12 are connected in series between the positive terminal of bus bar twelve and the negative terminal of bus bar nine. The first and ninth voltage supplies are left unconnected.

The flexibility of the voltage distributor in providing direct correspondence between the defect+free spans of the sheet material and the corresponding matrix bus bar for any possible defect pattern will be appreciated. Consecutive operation for the different spans is not required and partitions are simultaneously made in the FIG. 6 matrix. While the read-out from each of the cells of the FIG. 6 matrix may be separately recorded, the resulting read-out arrangement in the circuit of FIG. 7 remains of particular utility in providing a cumulative read-out \for all of the simultaneously derived partitions as a total for comparison with that for partition of a different section of sheet material having another defect pattern.

It will be appreciated that large bus bar spans will involve larger voltages, and all the resistors 75, 76 and 77 previously indicated as optionally included in the matrix of FIG. 6 play a helpful role. Thus, if the relays all have a common rating, the resistance of each cell is desirably proportional to the cell span to maintain equal voltages and equal currents in all of the cells in a given current path. With, for example, ten volt sources and relays having 1,000 oh-m coils with a rated 10 milliamperes operating current, the resistor 75 of each twospan cell 61 is of the order of 1,000 ohms to limit the current to the rated value at 20 volts. correspondingly, to maintain the rated conditions for the S-span and 4- span cells 62 and 63, the respective resistors 76 and 77 are 2,000 and 3,000 ohms.

(2) A parallel-type voltage distribut0r.Problems associated with the switching of high voltages are elirni inated by use of a parallel type O\f power supply in which the same given voltage is applied to each defect-free span of the matrix. Such a circuit is illustrated in FIG. SE in which defect relay contacts a, b, c and d are again horizontally alined with their respective defect relays -1 through 85-12 of FIG. 8B to clearly indicate the rearrangement of the same switching contacts for parallel operation. Voltage supplies 88-1 to 88-12 represent the twelve supplies for the respective twelve units of the sheet, material of FIG. 8A and the bus bar spans of the FIG. 6 matrix. As in FIG. 8C the positive and negative bus bar terminals are also shown.

The contact pairs a, b, c, -and d of relays 88-1 through 88-12 are connected differently in the parallel circuit of FIG. 8B than the series circuit of FIG. 8C in order to remove the common potential between bus bars effectively bounding the sides of a defect unit or strip of units. Thus each nth defect relay has its normally closed contact pair b connected between the positive terminals of voltage supplies n and n1 and its normally closed contact pair c connected between the negative terminals of the same supplies. The connection from the positive terminal of the n1 supply is made to the common pole of the contact pairs a and b" of the nth relay. The end terminal of normally open contact pair a is connected to the positive terminal of matrix bus bar 11-1. The negative terminal of supply n is connected to the common pole of normally closed contact pairs 0 and d. The end terminal of normally open contact pair d is connected to the negative terminal of matrix bus bar n. End connections for the first and last units are simple. Contact pairs a and b of the first defect relay 88-1 are merely left unconnected, and the end contact of normally closed pair 0 becomes the minus zero bus bar terminal and no zeropower 1'? supply is needed. The positive terminal of the twelfth voltage supply is the positive terminal of the twelfth bus bar.

In operation, in the absence of defects, the defect relay contact pairs a, b, c, d in their normal or unenergized condition connect all of the voltage sources in parallel. The source voltage is applied across the matrix from. the positive terminal of bus bar 12 to the zero bus bar. Under the defect conditions of FIG. 8A, and with the defect switches and defect relays of FIG. 8B correspondingly actuated, the voltage distribution efiected in the FIG. 8E circuit is shown by FIG. 8F. As in the previous case, the control voltage is applied, in correct polarity, between bus bars 8 and 1 and bus bars 12 and 9. The two distributed voltages are the same due to the parallel connection, and no voltage connections are made to the bus bars between which lie defect units one and eight.

With the parallel-connected voltage distributor, the cell relays of the matrix are chosen with wide-range operating characteristics based on a maximum coil current occurring upon application of the matrix voltage across one cell. The same source voltage may be applied to one relay or (in the example delineated by the FIG. 6 matrix) to as many as three relays in series. For larger matrices, the pickup range would be higher, calling for high relay sensitivity or operating current range. For such operation the cell resistors 75, 76 and 77 of FIG. 6 may be made equal to each other or omitted.

For either type of voltage distribution, it may be seen that voltages are applied through a switching system lending itself to automatic application to different spans of the same matrix. Reference to a constant or fixed bus bar becomes unnecessary and, in effect, voltage terminal bus bars chosen afresh for each partitioning operation.

D. Matrix universal cell blocks In what is here termed a universal cel-l block, a relatively simple matrix size switching circuit and small number of diodes can be switched to correctly interconnect the diodes across all the bus bar spans of a partitioning matrix for any selected size span corresponding to a selected one of a set of a integers. Such an arrangement offers maximum flexibility and versatility of the matrix for a given number of switches and diodes in terms of matrix rearrangement for different groups of a blocks and reassignment of a rank.

Looking now to FIG. 9, a universal cell block 89 incorporating a switching arrangement for a matrix having a maximum span of twelve is illustrated. Any a span from one to twelve is accommodated with no more than twelve cell or read-out relays and twelve diodes in the block. Larger span matrices are effectively utilized for fully realizing the advantages of the universal cell as well as solving more difiicult partitioning problems, the bus bars 90 comprising plus bus bars numbered 0, 1, 2, 3 12 providing a maximum matrix span equal to that of FIG. 6 and operable by the voltage distributors already shown in connection with FIGS. 8A to 8F.

Each cell of the FIG. 9 matrix (separately shown in FIG. 11), has a series-connected relay 91 and a diode 92. Each cell series combination has its upper terminal connected directly or permanently to one 'of the bus bars 1 to 12, and for aid in identification the relay coils in FIG. 9 also bear the number of the bus bar which carries them. The lower end of each cell is connected through a multiposition switching arrangement to any one of the lower numbered bus bars.

Twelve universal switching relays 93 are employed. As shown in FIG. the relay coils are connected across a control voltage source 94 through a l2-p'osition selector switch 95. The selector switch positions and the universal switching relay coils 93 connected to the respective switch positions are further numbered 1 through 12. The number of the selector switch position and the universal switching relay coil actuated thereby corresponds to the selected a integer or bus bar span rather than to the number of a cell or matrix bus bar. In the example shown, with the selector switch 95 set at its fourth position, the actuation of universal relay 93-4 interconnects each available span of four bus bars in the matrix with the appropriate ones of the 12 cell relays 91 required for the complete connection.

The relatively small total of switching contact pairs 93a of the universal switching relays 93-1 through 93-12 are enclosed within the broken line triangle in FIG. 9 bearing the reference number 93a. Each contact pair 93a Within the triangular configuration bears a further number 1 through 12 corresponding to the identification of the universal relay coil in FIG. 10. The number of switching contact pairs of each universal relay 93 is equal to the number of spans of that size, which is to say it is also equal to the number of cells which must be connected in that block. Universal relay 93-1 accordingly has 12 contact pairs 93a-1 respectively connected between the lower terminal of each cell and the bus bar immediately (span difference of one) below it. At the other extreme, since the number of universal cell contact pairs or switches required for a given span varies inversely with the size of the span, only one contact pair 91a-12 is required for a span of 12. There is, of course, only one cell, namely 91-12, which can possibly be connected across the 12-span matrix.

In operation, for any selected span or size, all useful spans of that number within the matrix are interconnected. With the exemplary span of four, the upper nine of the twelve cells are connected, and the cells 91-1, 2 and -3 are left open as desired. The arrangement also lends itself to economical expansion of the total matrix size, particularly for the more diflicult problems (i.e., difiicult by conventional partitioning calculations) in which larger integers make up the integer set and in which priority or rank is likely to be assigned to the higher integers of a set. Thus, since the integer 1 in a partitioning set would be useful in practical problem solving only where partitions could not be obtained with numbers other than 1, the facilities to accommodate the integer 1 and successively higher integers may be omitted in many cases, thus saving per progressive omission both one cell plus the switch set having the greatest required number of contacts. On the other hand, addition of a higher integer choice requires a relatively small number of switching contact pairs for the added cell involved.

Turning now to the rank and read-out means of the universal cell block of FIG. 9 for whatever a integer has been selected, each cell relay 91 has a normally closed contact pair 91a and a normally open contact pair 91b (also indicated in FIG. 11). The rank or priority switching contacts 91a are connected in the bus bar carrying that cell relay.

As shown in FIG. 9 each such normally closed contact pair 91a-1 to 9111-12 is connected to interrupt bus bar 1 to 12 respectively to the left of the connection of the cell to the bus bar. Current flow through any of the cell relays thus results in interrupting the voltage supply connections of the same-numbered bus bar to the next a, block. While in FIG. 9 the left hand ends of a bus bar are indicated with negative plurality as for return to the negative side of the voltage supply, it will be appreciated the next ranking a block may be inserted there instead.

A simple circuit for indicating the number of relays actuated in any selector switch position (which is the occurrence of the selected a, integer involved in a given partition) is shown in FIG. 10 in which a galvanometer 96 or other indicating or control instrument is connected across the control voltage source 94 through multiple possible paths. Each such path may suitably consist of a read-out resistor 97 and one of thenormally open contact pairs 918-1 through 91b-12 of the 12 bus bar cell relays. The current flow may thus be made to vary 

1. MEANS FOR DETERMINING THE PRESENCE OF A SELECTED ONE OF A CLASS OF GIVEN INTEGERS IN A PARTITION OF A GIVEN NUMBER WHICH COMPRISES A CONSECUTIVELY ORDERED ARRAY OF TERMINALS EQUAL TO SAID GIVEN NUMBER, DIODE SPANNING LINKS CONNECTED BETWEEN EACH SET OF TERMINALS SEPARATED BY A SPAN EQUAL TO ANY OF SAID CLASS OF INTEGERS, SAID LINKS BEING POLED FOR CONDUCTION FROM HIGHER TO LOWER ORDERED TERMINALS, MEANS FOR CONNECTING A VOLTAGE SOURCE BETWEEN TERMINALS SEPARATED BY A SPAN EQUAL TO SAID NUMBER TO PROVIDE A DIRECTIONAL CURRENT FLOW THROUGH ALL LINKED PATHS BETWEEN SAID TERMINALS, AND READOUT MEANS FOR DETECTING CURRENT FLOW THROUGH A DIODE LINK REPRESENTING THE SELECTED ONE OF SAID CLASS. 